Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12117 |
| automorphism | Aut(G) = #72308067 (order 2048) · Inn(G) = #1595 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1595 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | #2017 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
196 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,15)(4,20)(5,24)(6,13)(7,12)(8,14)(9,17)(10,22)(11,27)(16,44)(18,52)(19,43)(21,55)(23,40)(25,33)(26,36)(28,41)(29,37)(30,39)(31,38)(32,59)(34,50)(35,51)(42,46)(45,53)(47,64)(48,62)(49,60)(54,58)(56,61)(57,63);(1,3,8,19)(2,9,14,35)(4,16,23,48)(5,25,28,50)(6,18,7,31)(10,32,37,61)(11,38,39,52)(12,34,13,33)(15,41,43,24)(17,30,51,27)(20,53,40,60)(21,56,26,59)(22,46,29,47)(36,62,55,44)(42,63,64,54)(45,57,49,58);(1,4)(2,10)(3,16)(5,26)(6,29)(7,22)(8,23)(9,32)(11,36)(12,40)(13,20)(14,37)(15,42)(17,49)(18,47)(19,48)(21,28)(24,54)(25,59)(27,57)(30,58)(31,46)(33,53)(34,60)(35,61)(38,62)(39,55)(41,63)(43,64)(44,52)(45,51)(50,56)Polycyclic presentation:
2^8|b^2=h,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gh,d^b=df,e^b=egh,f^b=fh,d^c=dg,e^c=eh,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | #151152, #151153, #151156, … |
| maximal normal subgroup of | #151152, #151153, #151156, … |
| maximal quotient of | #123012, #123014, #123020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5106);